Oberwolfach Preprints (OWP)
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- ItemDiameter and Connectivity of Finite Simple Graphs II(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Hibi, Takayuki; Saeedi Madani, SaraLet $G$ be a finite simple non-complete connected graph on $[n] = \{1, \ldots, n\}$ and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. The final goal of this paper is to determine all sequences of integers $(n,f,d,k)$ with $n\geq 8$, $f\geq 0$, $d\geq 2$ and $k\geq 1$ for which there exists a finite simple non-complete connected graph on $[n]$ with $f=f(G)$, $d=\mathrm{diam}(G)$ and $k=\kappa(G)$.
- ItemLocal Existence and Conditional Regularity for the Navier-Stokes-Fourier System Driven by Inhomogeneous Boundary Conditions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Abbatiello, Anna; Basarić, Danica; Chaudhuri, Nilasis; Feireisl, EduardWe consider the Navier–Stokes–Fourier system with general inhomogeneous Dirichlet–Neumann boundary conditions. We propose a new approach to the local well-posedness problem based on conditional regularity estimates. By conditional regularity we mean that any strong solution belonging to a suitable class remains regular as long as its amplitude remains bounded. The result holds for general Dirichlet-Neumann boundary conditions provided the material derivative of the velocity field vanishes on the boundary of the physical domain. As a corollary of this result we obtain: Blow up criteria for strong solutions; Local existence of strong solutions in the optimal Lp - Lq framework; Alternative proof of the existing results on local well posedness.
- ItemThe Alternating Halpern-Mann Iteration for Families of Maps(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Firmino, Paulo; Pinto, PedroWe generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature.
- ItemOn Overgroups of Distinguished Unipotent Elements in Reductive Groups and Finite Groups of Lie Type(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Bate, Michael; Böhm, Sören; Martin, Benjamin; Röhrle, GerhardSuppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good A1 subgroups of G, introduced by Seitz. We also formulate a counterpart of Korhonen's theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen's theorem for Lie algebras.
- ItemProof Mining and the Convex Feasibility Problem : the Curious Case of Dykstra's Algorithm(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Pinto, PedroIn a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector's bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining.